Cocommutative Hopf Algebra Research Articles

Yang–Baxter operators in symmetric categories

ABSTRACTWe introduce non-degenerate solutions of the Yang–Baxter equation in the setting of symmetric monoidal categories. Our theory includes non-degenerate set-theoretical solutions as basic examples. However, infinite families of non-degenerate solutions (that are not of set-theoretical type) appear. As in the classical theory of Etingof, Schedler, and Soloviev, non-degenerate solutions are classified in terms of invertible 1-cocycles. Braces and matched pairs of cocommutative Hopf algebras (or braiding operators) are also generalized to the context of symmetric monoidal categories and turn out to be equivalent to invertible 1-cocycles.

Character varieties as a tensor product

In this short note we show that representation and character varieties of discrete groups can be viewed as tensor products of suitable functors over the PROP of cocommutative Hopf algebras. Such view point has several interesting applications. First, it gives a straightforward way of deriving the functor sending a discrete group to the functions on its representation variety, which leads to representation homology. Second, using a suitable deformation of the functors involved in this construction, one can obtain deformations of the representation and character varieties for the fundamental groups of 3-manifolds, and could lead to better understanding of quantum representations of mapping class groups.

Hopf algebras and invariants of the Johnson cokernel

We show that if H is a cocommutative Hopf algebra, then there is a natural action of Aut(F_n) on the nth tensor power of H which induces an Out(F_n) action on a quotient \overline{H^{\otimes n}}. In the case when H=T(V) is the tensor algebra, we show that the invariant Tr^C of the cokernel of the Johnson homomorphism studied in [J. Conant, The Johnson cokernel and the Enomoto-Satoh invariant, Algebraic and Geometric Topology, 15 (2015), no. 2, 801--821.] projects to take values in the top dimensional cohomology of Out(F_n) with coefficients in \overline{H^{\otimes n}}. We analyze the n=2 case, getting large families of obstructions generalizing the abelianization obstructions of [J. Conant, M. Kassabov, K. Vogtmann, Higher hairy graph homology, Journal of Topology, Geom. Dedicata 176 (2015), 345--374.].

The Hochschild complex of a twisting cochain

Given any twisting cochain t:C→A, where C is a connected, coaugmented chain coalgebra and A is an augmented chain algebra over an arbitrary commutative ring R, we construct a twisted extension of chain complexes [Display omitted] of which both the well-known Hochschild complex of an augmented, associative algebra and the coHochschild complex of a coaugmented, coassociative coalgebra [13] are special cases. We therefore call H(t) the Hochschild complex of the twisting cochain t.We explore the extent of the naturality of the Hochschild complex construction and apply the results of this exploration to determining conditions under which H(t) admits multiplicative or comultiplicative structure. In particular, we show that the Hochschild complex on a chain Hopf algebra always admits a natural comultiplication.Furthermore, when A is a chain Hopf algebra, we determine conditions under which H(t) admits an rth-power map extending the usual rth-power map on A and lifting the identity on C. As special cases, we obtain that both the Hochschild complex of any cocommutative Hopf algebra and the coHochschild complex of the normalized chain complex of a simplicial double suspension admit power maps.

Graded Left Symmetric Pseudo-algebras

In the present article, we introduce G-graded left symmetric H-pseudoalgebras, where G is a grading group, and H is a cocommutative Hopf algebra. Some results about associative H-pseudoalgebras in [23] are generalized. The commutator algebras of the G-graded left symmetric H-pseudo-algebras are Lie H-pseudoalgebras, which are classified when the grading group is trivial in [3]. We investigate the left symmetric structure of Lie H-pseudoalgebras W(𝔟), S(𝔟), and He defined in [3].

An Example in the Singer Category of Algebras with Coproducts at Odd Primes

In 2005, William M. Singer introduced the notion of k-algebra with coproducts for any commutative ring k and showed that the algebra of operations on the cohomology ring of any cocommutative \(\mathbb {F}_{2}\)-Hopf algebra can be endowed with such structure. In this paper, we show that the same is true when the ground field of the cocommutative Hopf algebra is \(\mathbb {F}_{p}\), p is any odd prime, and the algebra of operations \({\mathcal {B}}(p)\) is equipped with an exotic coproduct. We also give an explicit description of the coalgebra with products dual to \({\mathcal {B}} (p)\).

A Torsion Theory in the Category of Cocommutative Hopf Algebras

The purpose of this article is to prove that the category of cocommutative Hopf $K$-algebras, over a field $K$ of characteristic zero, is a semi-abelian category. Moreover, we show that this category is action representable, and that it contains a torsion theory whose torsion-free and torsion parts are given by the category of groups and by the category of Lie $K$-algebras, respectively.

Leibniz H-pseudoalgebras

We define a Leibniz H-pseudoalgebra over a cocommutative Hopf algebra H, and a tensor product of Leibniz H-pseudoalgebras, and prove that every Leibniz H-pseudoalgebra of free rank one over H is a Lie H-pseudoalgebra. To characterize simple Leibniz H-pseudoalgebras, we define the Leibniz cohomology of Leibniz H-pseudoalgebras and determine some simple Leibniz H-pseudoalgebras by calculating Leibniz cohomological groups of finite simple Lie H-pseudoalgebras defined in [2].

The monomial basis and the $$Q$$ Q -basis of the Hopf algebra of parking functions

Consider the vector space $\mathbb{K}\mathcal{P}$ spanned by parking functions. By representing parking functions as labeled digraphs, Hivert, Novelli and Thibon constructed a cocommutative Hopf algebra PQSym$^{*}$ on $\mathbb{K}\mathcal{P}$. The product and coproduct of PQSym$^{*}$ are analogous to the product and coproduct of the Hopf algebra NCSym of symmetric functions in noncommuting variables defined in terms of the power sum basis. In this paper, we view a parking function as a word. We shall construct a Hopf algebra PFSym on $\mathbb{K}\mathcal{P}$ with a formal basis $\{M_a\}$ analogous to the monomial basis of NCSym. By introducing a partial order on parking functions, we transform the basis $\{M_a\}$ to another basis $\{Q_a\}$ via the M\"{o}bius inversion. We prove the freeness of PFSym by finding two free generating sets in terms of the $M$-basis and the $Q$-basis, and we show that PFSym is isomorphic to the Hopf algebra PQSym$^{*}$. It turns out that our construction, when restricted to permutations and non-increasing parking functions, leads to a new way to approach the Grossman-Larson Hopf algebras of ordered trees and heap-ordered trees.

Properties of semisimple Hopf algebras

Semisimple finite-dimensional Hopf algebras all of whose non-one-dimensional irreducible modules of the same dimension are isomorphic are considered. The one-dimensional submodules of the tensor squares of irreduciblemodules are indicated and the almost cocommutative Hopf algebras and the left ideal coideals are described.

Piecewise Principal Coactions of Co-Commutative Hopf Algebras

Principal comodule algebras can be thought of as objects representing principal bundles in non-commutative geometry. A crucial component of a principal comodule algebra is a strong connection map. For some applications it suffices to prove that such a map exists, but for others, such as computing the associated bundle projectors or Chern-Galois characters, an explicit formula for a strong connection is necessary. It has been known for some time how to construct a strong connection map on a multi-pullback comodule algebra from strong connections on multi-pullback components, but the known explicit general formula is unwieldy. In this paper we derive a much easier to use strong connection formula, which is not, however, completely general, but is applicable only in the case when a Hopf algebra is co-commutative. Because certain linear splittings of projections in multi-pullback comodule algebras play a crucial role in our construction, we also devote a significant part of the paper to the problem of existence and explicit formulas for such splittings. Finally, we show example application of our work.

Locality Criteria for Cocommutative Hopf Algebras

We prove that a finite-dimensional cocommutative Hopf algebra H is local, if and only if the subalgebra generated by the first term of its coradical filtration H 1 is local. In particular if H is connected, H is local if and only if all the primitive elements of H are nilpotent.

The Dixmier-Moeglin Equivalence for Cocommutative Hopf Algebras of Finite Gelfand-Kirillov Dimension

Let k be an algebraically closed field of characteristic zero and let H be a noetherian cocommutative Hopf algebra over k. We show that if H has polynomially bounded growth then H satisfies the Dixmier-Moeglin equivalence. That is, for every prime ideal P in Spec(H) we have the equivalences $$P\; \text{primitive}\iff P\; \text{rational}\iff P\; \text{locally closed in}~\text{Spec}(H).$$ We observe that examples due to Lorenz show that this does not hold without the hypothesis that H have polynomially bounded growth. We conjecture, more generally, that the Dixmier-Moeglin equivalence holds for all finitely generated complex noetherian Hopf algebras of polynomially bounded growth.

Crossed products over weak Hopf algebras related to cleft extensions and cohomology

The authors present the general theory of cleft extensions for a cocommutative weak Hopf algebra H. For a right H-comodule algebra, they obtain a bijective correspondence between the isomorphisms classes of H-cleft extensions AH ↪ A, where AH is the subalgebra of coinvariants, and the equivalence classes of crossed systems for H over AH. Finally, they establish a bijection between the set of equivalence classes of crossed systems with a fixed weak H-module algebra structure and the second cohomology group \(H_{\phi _{Z(A_H )} }^2 \) (H,Z(AH)), where Z(AH) is the center of AH.

Koszul duality theory for operads over Hopf algebras

The transfer of the generating operations of an algebra to a homotopy equivalent chain complex produces higher operations. The first goal of this paper is to describe precisely the higher structure obtained when the unary operations commute with the contracting homotopy. To solve this problem, we develop the Koszul duality theory of operads in the category of modules over a cocommutative Hopf algebra. This gives rise to a simpler category of homotopy algebras and infinity-morphisms, which allows us to get a new description of the homotopy category of algebras over such operads. The main example of this theory is given by Batalin-Vilkovisky algebras.

Cohomology of algebras over weak Hopf algebras

In this paper we present the Sweedler cohomology for a cocommutative weak Hopf algebra H. We show that the second cohomology group classifies completely the weak crossed products, having a common preunit, of H with a commutative left H-module algebra A.

The Equivariant Brauer Group of a Cocommutative Hopf Algebra

We investigate the Brauer group BRM(k, H) of a cocommutative Hopf algebra H (H can be nonfinitely generated). BRM(k, H) is the bigger Brauer group of H-module Taylor–Azumaya algebras (which do not necessarily contain a unit). An exact sequence is obtained. Furthermore, we prove the surjectivity of under certain circumstances. This generalizes Beattie's exact sequence to the “infinite” case.

THE CALABI–YAU PROPERTY OF TWISTED SMASH PRODUCTS

Let H be a finite-dimensional cocommutative semisimple Hopf algebra and A * H a twisted smash product. The Calabi–Yau (CY) property of twisted smash product is discussed. It is shown that if A is a CY algebra of dimension dA, a necessary and sufficient condition for A * H to be a CY Hopf algebra is given.

Quadratic Differential Operators, Bicharacters and • Products

For a commutative cocommutative Hopf algebra, we study the relationship between a certain linear map defined via a bicharacter, an exponential of a quadratic differential operator, and a • product obtained via twisting by a bicharacter. This new relationship between • products and exponentials of quadratic differential operators was inspired by studying the exponential of a quadratic differential operator introduced in [7] and used in the theory of twisted modules of lattice vertex algebras.

Lie algebra structures on cohomology complexes of some H-pseudoalgebras

Suppose H is a cocommutative Hopf algebra and P is the operad Ass, or Lie. For any left H-module V, we construct a graded Lie algebra (LP(V)=⨁n∈ZLPn(V),[,]), and prove that φ∈LP1 satisfies [φ,φ]=0 if and only if it determines a pseudoalgebra structure on V over the operad P, and LP(V) is a differential graded Lie algebra with a differentiation determined by the structure map φ of the H-pseudoalgebra V. Moreover, we construct a Lie algebra by using symmetric Schouten product for any pseudoalgebra V over the operad Lie.